Problem: Simplify the following expression and state the condition under which the simplification is valid. You can assume that $q \neq 0$. $y = \dfrac{8}{8(4q - 9)} \div \dfrac{-8}{4(4q - 9)} $
Dividing by an expression is the same as multiplying by its inverse. $y = \dfrac{8}{8(4q - 9)} \times \dfrac{4(4q - 9)}{-8} $ When multiplying fractions, we multiply the numerators and the denominators. $y = \dfrac{ 8 \times 4(4q - 9) } { 8(4q - 9) \times -8 } $ $ y = \dfrac{32(4q - 9)}{-64(4q - 9)} $ We can cancel the $4q - 9$ so long as $4q - 9 \neq 0$ Therefore $q \neq \dfrac{9}{4}$ $y = \dfrac{32 \cancel{(4q - 9})}{-64 \cancel{(4q - 9)}} = -\dfrac{32}{64} = -\dfrac{1}{2} $